In mathematics, van der Corput's method generates estimates for exponential sums. The method applies two processes, the van der Corput processes A and B which relate the sums into simpler sums which are easier to estimate.
The processes apply to exponential sums of the form
where f is a sufficiently smooth function and e(x) denotes exp(2πix).
Process Aedit
To apply process A, write the first difference fh(x) for f(x+h)−f(x).
Assume there is H ≤ b−a such that
Then
Process Bedit
Process B transforms the sum involving f into one involving a function g defined in terms of the derivative of f. Suppose that f' is monotone increasing with f'(a) = α, f'(b) = β. Then f' is invertible on [α,β] with inverse u say. Further suppose f'' ≥ λ > 0. Write
We have
Applying Process B again to the sum involving g returns to the sum over f and so yields no further information.
Exponent pairsedit
The method of exponent pairs gives a class of estimates for functions with a particular smoothness property. Fix parameters N,R,T,s,δ. We consider functions f defined on an interval [N,2N] which are R times continuously differentiable, satisfying
uniformly on [a,b] for 0 ≤ r < R.
We say that a pair of real numbers (k,l) with 0 ≤ k ≤ 1/2 ≤ l ≤ 1 is an exponent pair if for each σ > 0 there exists δ and R depending on k,l,σ such that
uniformly in f.
By Process A we find that if (k,l) is an exponent pair then so is .
By Process B we find that so is .
A trivial bound shows that (0,1) is an exponent pair.
The exponent pair conjecture states that for all ε > 0, the pair (ε,1/2+ε) is an exponent pair. This conjecture implies the Lindelöf hypothesis.
Referencesedit
Ivić, Aleksandar (1985). The Riemann zeta-function. The theory of the Riemann zeta-function with applications. New York etc.: John Wiley & Sons. ISBN 0-471-80634-X. Zbl 0556.10026.
Montgomery, Hugh L. (1994). Ten lectures on the interface between analytic number theory and harmonic analysis. Regional Conference Series in Mathematics. Vol. 84. Providence, RI: American Mathematical Society. ISBN 0-8218-0737-4. Zbl 0814.11001.
Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, eds. (2006). Handbook of number theory I. Dordrecht: Springer-Verlag. ISBN 1-4020-4215-9. Zbl 1151.11300.